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Contests & Programs
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Collinear points
tenplusten 2
N
3 hours ago
by Blackbeam999
Let
be three collinear points and
three other collinear points. Let
be the intersection of the lines
and
,respectively. If
and
.Prove that 
I hope you will use Pappus theorem in your solutions.









I hope you will use Pappus theorem in your solutions.
2 replies
Simple Geometry
AbdulWaheed 0
3 hours ago
Source: EGMO
Try to avoid Directed angles
Let ABC be an acute triangle inscribed in circle
. Let
be the midpoint of the arc
not containing
and define
similarly. Show that the orthocenter of
is the incenter
of
.
Let ABC be an acute triangle inscribed in circle








0 replies
IMO Shortlist 2014 G6
hajimbrak 30
N
Today at 12:01 AM
by awesomeming327.
Let
be a fixed acute-angled triangle. Consider some points
and
lying on the sides
and
, respectively, and let
be the midpoint of
. Let the perpendicular bisector of
intersect the line
at
, and let the perpendicular bisector of
intersect the lines
and
at
and
, respectively. We call the pair
, if the quadrilateral
is cyclic.
Suppose that the pairs
and
are interesting. Prove that 
Proposed by Ali Zamani, Iran


















Suppose that the pairs



Proposed by Ali Zamani, Iran
30 replies
60^o angle wanted, equilateral on a square
parmenides51 4
N
Yesterday at 11:49 PM
by MathIQ.
Source: 2019 Austrian Mathematical Olympiad Junior Regional Competition , Problem 2
A square
is given. Over the side
draw an equilateral triangle
on the outside. The midpoint of the segment
is
and the midpoint of the side
is
. Prove that
.
.
(Karl Czakler)








.
(Karl Czakler)
4 replies
Serbian selection contest for the IMO 2025 - P2
OgnjenTesic 9
N
Yesterday at 11:47 PM
by hectorleo123
Source: Serbian selection contest for the IMO 2025
Let
be an acute triangle. Let
be the reflection of point
over the line
. Let
and
be the circumcenter and the orthocenter of triangle
, respectively, and let
be the midpoint of segment
. Let
and
be the points where the reflection of line
with respect to line
intersects the circumcircle of triangle
, where point
lies on the arc
not containing
. If
is a point on the line
such that
, prove that
.
Proposed by Strahinja Gvozdić





















Proposed by Strahinja Gvozdić
9 replies
collinear wanted, regular hexagon
parmenides51 3
N
Yesterday at 11:34 PM
by MathIQ.
Source: 2023 Austrian Mathematical Olympiad , Junior Regional Competition , Problem 2
Let
be a regular hexagon with sidelength s. The points
and
are on the diagonals
and
, respectively, such that
. Prove that the three points
,
and
are on a line.
(Walther Janous)









(Walther Janous)
3 replies
Geometric inequality with angles
Amir Hossein 7
N
Yesterday at 11:06 PM
by MathIQ.
Let
, and
be the angles of a triangle, and let
, and
. If
, show that
![\[s(s - a)(s - b)(s -c) \geq 0.\]](//latex.artofproblemsolving.com/1/3/4/1344a0cbbd4e1aa4bcf6a1e777501c62daff32b3.png)
When does equality hold?





![\[s(s - a)(s - b)(s -c) \geq 0.\]](http://latex.artofproblemsolving.com/1/3/4/1344a0cbbd4e1aa4bcf6a1e777501c62daff32b3.png)
When does equality hold?
7 replies
IMO 2014 Problem 3
v_Enhance 103
N
Yesterday at 10:59 PM
by Mysteriouxxx
Source: 0
Convex quadrilateral
has
. Point
is the foot of the perpendicular from
to
. Points
and
lie on sides
and
, respectively, such that
lies inside triangle
and
Prove that line
is tangent to the circumcircle of triangle
.











![\[
\angle CHS - \angle CSB = 90^{\circ}, \quad \angle THC - \angle DTC = 90^{\circ}. \]](http://latex.artofproblemsolving.com/f/8/c/f8c35912c531eb2dae8f62b74666b7757d2b9ba3.png)


103 replies
three discs of radius 1 cannot cover entirely a square surface of side 2
parmenides51 1
N
Yesterday at 8:17 PM
by Blast_S1
Source: 2014 Romania NMO VIII p4
Prove that three discs of radius
cannot cover entirely a square surface of side
, but they can cover more than
of it.



1 reply
2025 Caucasus MO Juniors P6
BR1F1SZ 2
N
Yesterday at 7:38 PM
by IEatProblemsForBreakfast
Source: Caucasus MO
A point
is chosen inside a convex quadrilateral
. Could it happen that



2 replies
